Note that in order to control \(Q^{+}_{\Phi _{s},b^{\infty }}\) , the small relative velocities, the kinetic potential \(\Phi (u)\)  needs to vanish as \(|u|\rightarrow 0\) . This is not the case for the Maxwell Molecules case. Indeed, Maxwell molecules is a critical case for uniform propagation of \(L^{r}\) -integrability for general initial data in \(L^{r}\)  in the sense that as soon as \(\gamma <0\) , i.e. soft potential case, the uniform propagation of moments is lost for general initial data as discussed in [1]} let alone the \(L^{r}_{\ell }\) -norms For the non-cutoff Boltzmann problem this is not the case and the analysis is quite different.. In order to compensate for this criticality issue, we will use the additional compactness gained through propagation of entropy after noticing that we implicitly have the additional requirement on the initial data
\(\int _{\mathbb {R}^{d}}f_0(v)\ln \big (f_0(v)\big )dv<\infty \,.\) 
